
Strongly compact cardinals and SCH
Narusevych, Mykyta ; Šaroch, Jan (advisor) ; Krajíček, Jan (referee)
The thesis is devoted to the cardinal arithmetic. The first step is to formulate the Singular Cardinals Hypothesis (SCH) which simplifies the cardinal exponentiation of sin gular cardinal numbers. We then define stationary sets and closed and unbounded subsets of an ordinal number. The main goal is to prove the Silver's theorem and the corollary which states, that if SCH holds for all singular cardinals with countable cofinality, then it holds everywhere. In the last chapter we define strongly compact cardinal numbers and prove some of their properties. Finally, we prove the Solovay's theorem, which states that SCH holds everywhere above a strongly compact cardinal. 1


Rings with restricted minimum condition
Krasula, Dominik ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Ring is artintian if and only if all of its factors are artinian. We say that ring R satisfies the restricted minimum condition, if for every essenctial ideal, corresponding factor ring is artinian. We will call such ring RM ring for short. Similarly as the class of artinian rings, the class of RM rings is closed under fac tors and finite direct products. In this thesis we prove that restricted minimum condition is satisfied in coordinate rings, ring (R × R)[x] and noetherian CDR domains. We investigate the relation between unique factorization domains and RM domains. In last chpater, we will focus are attention to polynomial rings, proving that if ring R[x] is RM then R is semisimple. Laurents polynomials over domain R are RM rings if and only if R is a field. 1


Structure of pureinjective abelian groups
Jankovec, Filip ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the structure of the pureinjective abelian groups. We de scribe some equivalent characterizations of the pureinjective modules. Furthermore, we thoroughly discuss the special case of the pureinjective modules over principal ideal do main. We show that every pureinjective abelian group can be written unambiguously only using cyclic groups, Prüfer groups and the group of rational numbers. Moreover, an abelian group can be written in this form if and only if it is a pureinjective abelian group. 1


Diamond principles and GCH
Fuková, Kateřina ; Šaroch, Jan (advisor) ; Chodounský, David (referee)
Diamond principles and the generalized continuum hypothesis are assertions related to infinite combinatorics. This thesis studies various connections between these assertions. From numerous formulations of diamond principles, it explicitly mentions exactly two of them: ♢S and ♢∗ S. Apart from an overview of the basic notions involved in this study, the thesis also contains a concise proof of Shelah's theorem published in the paper "Diamonds" in 2010. 1


Spectrum problem
Ježil, Ondřej ; Krajíček, Jan (advisor) ; Šaroch, Jan (referee)
We study spectra of firstorder sentences. After providing some interesting examples of spectra we show that the class of spectra is closed under some simple settheoretic and algebraic operations. We then define a new class of definable operations generalizing the earlier constructions. Our main result is that the class of these operations is, in a suitable technical sense, closed under a form of iteration. This in conjunction with Cobham's characterisation of FP offers a new proof of Fagin's theorem and also of the JonesSelman characterisation of spectra as NE sets. 1

 

Testing the projectivity of modules
Matoušek, Cyril ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the problem of the existence of test modules for the projectivity. A right Rmodule is said to be a test module if it holds for every right Rmodule M that M is projective whenever T ∈ M⊥ . We show that test modules exist over right perfect rings, although their existence is not provable in ZFC in case of nonright perfect rings. In order to prove this, we use Shelah's uni formization principle, which is independent of the axioms of ZFC. Furthermore, we show that test modules exist over rings of finite global dimension assuming the weak diamond principle, which is also independent of ZFC. 1


Primes in discretely ordered quasiEuclidean domains
Sgallová, Ester ; Šaroch, Jan (advisor) ; Glivická, Jana (referee)
This thesis studies discretely ordered quasiEuclidean domains. The goal is to study primes and prime pairs in them and to answer the question, whether there can be a cofinal set of them. The first construction gives a domain that does not have a cofinal set primes. Another construction builds a principal ideal domain, which has a cofinal set of primes, but no two distinct nonstandard primes differ by a natural number, so there is not a cofinal set of prime pairs. Furthermore, the thesis describes a construction of a principal ideal domain, whitch has a cofinal set of prime apairs for any even positive integer a. 1


Max rings
Beneš, Daniel ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Topic of this thesis is max rings, which are the rings, whose nonzero modu les have maximal submodules. At the begining we prove a characterization of commutative max rings as rings with Tnilpotent Jacobson radical and von Ne umann regular factor ring of the Jacobson radical. Our next concern are group rings, where we describe all commutative group rings, that are max. These are the group rings, that are composed from a commutative max ring and an abelian torsion group, where is finitely many elements of order pn for p not invertible in the ring. Finally we use this characterization to construct noncommutative group rings, which are max but not perfect.


Multilinear Maps Over the Integers
Havránek, František ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
The thesis aims to describe the [CLT15] scheme, which is based on the Diffie Hellman scheme and uses multilinear maps over integers. This scheme enables an exchange of a key among several participants. The level κ scheme (using a κlinear map) enables the exchange of a key among κ + 1 participants. The thesis introduces the basic terms, describes the needed theory, the base of which is the Chinese Remainder Theorem, and also the preparation and usage of the scheme. The correctness of the scheme is proved as well and the related requirements on the basic parameters are discussed.
